20.4 Non Integrable Functions

Are there functions that are not Riemann integrable?

Yes there are, and you must beware of assuming that a function is integrable without looking at it.

The simplest examples of non-integrable functions are: in the interval [0, b]; and in any interval containing 0. These are intrinsically not integrable, because the area that their integral would represent is infinite.

There are others as well, for which integrability fails because the integrand jumps around too much.

An extreme example of this is the function that is 1 on any rational number and 0 elsewhere.

Thus the area chosen to represent a single slice in a Riemann sum will be either its width or 0 depending upon whether we pick a rational x or not at which to evaluate our integrand in that interval.

For this function no matter how small the intervals are, you can have a Riemann sum of 0 or of b - a.

In this case it is possible to use a cleverer definition of the area to define it. (You can argue, in essence, that there are so many more irrational points than rational ones, you can ignore the latter, and the integral will be 0.)

If we consider the area under the curve defined by in an interval between -a and b for positive a and b, the area has an infinite positive part between 0 and b and an infinite negative part between -a and 0. It is possible to define the area here so that these cancel out and meaning can be given to the net area. (If you leave out the interval between -d and d for any small d, the remaining area is finite, and can be computed. You can then take the limit of this area as d goes to 0. The result is called the principle part of the integral and can be and is so defined for functions like whose infinite areas can have opposite signs and can counterbalance each other.)